Police have said that they are pursuing all lines of inquiry, including that the murder might have been the result of a botched break-in.

On Friday, detectives appealed to the public to report any sightings of a person with “bloodstained clothes”.

But after it was revealed Miss Graves was attacked in her bedroom – without police stating whether or not there were any signs of forced entry – speculation has grown she may have been the victim of a targeted killing, possibly by someone close to her. Some relatives have had property taken by police for forensic analysis.

The murder took place at a £1.6 million house on Smugglers’ Lane. It belongs to old friends of Miss Graves’s family, Malcolm and Caroline Chamberlain (also known as Jane), who were spending the festive period with their twin children in Costa Rica.

Valerie Graves (PA)

Mr Chamberlain, 66, a wealthy company director and a former British Airways employee, designed the home himself on a nautical theme. On the waterfront, it has a swimming pool, a balcony in the shape of a ship’s bow, and extensive views.

It emerged after the killing that an account in Mr Chamberlain’s name appears on a swingers’ website. Police have reportedly been investigating an advert on the site, suggesting the property could be used as a venue for “casual sex” and a “discreet relationship”.

The profile contains a picture of the bald and stocky Mr Chamberlain posing naked in a rock pool, and another in which he appears to be brushing his teeth in the nude.

On the site, Mr Chamberlain is described as a “director of fun and pleasuring” who is looking for “one-on-one sex, threesome”, with a particular interest in “bondage and discipline, high heels”. Under the “income” field, it simply says, “Ok, I’m VERY wealthy”.

Mr Chamberlain denies creating the account. “He doesn’t understand how [the photos] got onto the internet,” said Paul Dixon, his lawyer. “They are private photos. He has not put them on the internet, which makes you question what has happened.”

Mr Chamberlain has been linked to sordidness before. In 1988, he was suspended from his job at Kingston council when two players from a basketball team he had coached were photographed for a magazine with porn actress Pamela Meadows at a leisure centre Mr Chamberlain was managing. He was later cleared of wrongdoing.

This may sound like a television thriller. In fact, the village of Bosham has been a location in not one but two popular crime dramas: Midsomer Murders and Poirot.

The Midsomer Murders episode, Written in Blood, was broadcast in 1998. The fictional victim, a middle-aged writer, was bludgeoned to death with a candlestick in his bedroom on an unseasonably stormy night, while indulging a fetish for women’s clothing.

In the Poirot programme, The Adventure of the Italian Nobleman, from 1993, an Italian aristocrat was also bludgeoned to death, but with a statue, by a man who was coincidentally called Mr Graves.

The perpetrators of both fictional murders were close to their victims. In the real-life investigation, however, detectives have not ruled out the possibility that Miss Graves was killed by a stranger.

Police have not revealed if there was any evidence of a break-in, or that any valuables were taken. This has fuelled rumours the murder was the result of a grudge.

Christmas should have been a particularly happy time for Miss Graves, who made felt animal figures. Not only was Dec 25 her birthday, but her mother, Eileen, 87 – who lives in Chichester with her 90-year-old brother Ron – had just left hospital after suffering from pneumonia.

Miss Graves had moved down from Scotland to become her mother’s live-in carer two months before. Now her mother was able to join her at the house in Bosham, together with friends and other family members, while Ron visited his son in London.

Although Miss Graves was in Bosham for most of the festive period, she left to spend Christmas day with her two children and two grandchildren in the capital.

Ernie Mears, 80, a friend of the family who visited the Bosham house on Boxing Day, said Miss Graves’s temporary absence had adversely affected the mood in the house.

“Our gathering this year was not as happy as it normally was because Valerie and her children weren’t there,” he said. “This year, Boxing Day there just wasn’t the same atmosphere.”

It is not known if Miss Graves’s Christmas departure was due to discord within the family. But in the absence of arrests, everyone who was in the house, as well as former associates of Miss Graves, has fallen under suspicion. According to Det Supt Nick May of Sussex police, officers are “trying to understand about Valerie’s life, who she associates with and who may be connected in that way”.

Police and Forensic Officers continue to search the property where the body of Valerie Graves was found (CHRISTOPHER PLEDGER)

Many of Miss Graves’s relatives’ mobile phones have been taken by police for analysis. Officers have also towed away a 14-foot motorboat that was moored near the property, and a silver Lexus SUV with a personalised number plate.

It is believed the Lexus belongs to Nigel Acres, 59, an architect who is the partner of Miss Graves’s older sister, Jan, 60. He was at the property at the time of the murder.

Over six feet tall, with curling grey hair and glasses, Mr Acres is an old friend of Mr Chamberlain. Both played for the Kingston basketball club in the Eighties; about 13 years ago, Mr Acres and his wife bought their current home in Weybridge, Surrey, from the Chamberlains.

Police have also informally interviewed former roadie Roy Wood, 65, who is the father of Miss Graves’s two children. A keen scuba diver, with long hair and weather-beaten face, he plays guitar in a Brighton-based rock band called That Legendary Wooden Lion, which he founded in the Seventies. He prefers to use the “alter-ego” Roy Weard.

Mr Wood, who has since married and is believed to be writing memoirs of his rock’n’roll past, has posted black-and-white pictures of himself on Facebook posing in a graveyard dressed as the vampire from Nosferatu, a 1922 cult German horror film.

On Friday, Miss Graves’s children, Tim Wood and Jemima Harrison, appealed to the public to help find her killer.

“She was a free spirit who enjoyed her life and was a talented artist,” said Mr Wood. “My mum was much loved and will be sorely missed by the whole family.”

Since the crime, local people have gathered outside the property, some shaking their heads in disbelief.

“It could have been a burglary that went wrong, but we don’t get many burglaries here,” said one elderly Bosham neighbour. “We just don’t get this sort of trouble at Smugglers Lane.”